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May 2017 Automorphism groups of Gaussian Bayesian networks
Jan Draisma, Piotr Zwiernik
Bernoulli 23(2): 1102-1129 (May 2017). DOI: 10.3150/15-BEJ771


In this paper, we extend earlier work on groups acting on Gaussian graphical models to Gaussian Bayesian networks and more general Gaussian models defined by chain graphs with no induced subgraphs of the form $i\to j-k$. We fully characterise the maximal group of linear transformations which stabilises a given model and we provide basic statistical applications of this result. This includes equivariant estimation, maximal invariants for hypothesis testing and robustness. In our proof, we derive simple necessary and sufficient conditions on vanishing subminors of the concentration matrix in the model. The computation of the group requires finding the essential graph. However, by applying Stúdeny’s theory of imsets, we show that computations for DAGs can be performed efficiently without building the essential graph.


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Jan Draisma. Piotr Zwiernik. "Automorphism groups of Gaussian Bayesian networks." Bernoulli 23 (2) 1102 - 1129, May 2017.


Received: 1 January 2015; Revised: 1 September 2015; Published: May 2017
First available in Project Euclid: 4 February 2017

zbMATH: 1381.62223
MathSciNet: MR3606761
Digital Object Identifier: 10.3150/15-BEJ771

Keywords: chain graphs , equivariant estimator , Gaussian graphical models , group action , invariant test , transformation family

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability


Vol.23 • No. 2 • May 2017
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